Strong Law of Large Numbers Application

This was a test question from last semester that I stared at blankly for a long time and couldn't figure out where to begin. Obviously since I need to show convergence almost surely, its the Strong Law of Large Numbers, but beyond that I got really stuck. I'd love some input on at least how to get started. Thanks!

Problem:

Deﬁne the sequence inductively by setting , and selecting randomly and uniformly from the interval . Prove that converges almost surely to a constant, and evaluate the limit.

Hint given (I'm having trouble with the /sum so I wrote out the sum expanded):

let

Re: Strong Law of Large Numbers Application

I don't see a SLLN here.

Why don't you obtain the distribution of and then take the log of that rv?

Re: Strong Law of Large Numbers Application

Re: Strong Law of Large Numbers Application

Re: Strong Law of Large Numbers Application

Quote:

Originally Posted by

**chisigma** If the answer is 'yes', then the quantity

is the solution of the difference equation...

(1)

... so that is...

(2)

... and setting is...

(1)

... so that is...

(2)

Kind regards

Re: Strong Law of Large Numbers Application

Thanks everyone for your help. Sorry I took so long to respond... the assignment is already turned in, but your comments have helped me understand this much better! Thank you